Re: horizontal composition

Re: horizontal composition

[jean benabou]

  I thought I had invented bicategories in 1967, and that, at the very 
beginning of the paper, in §1, I had defined the two composition laws 
and drawn pictures to explain them. Of course I denoted by capital 
letters the 1-cells, thinking of functors, and by small letters the 
2-cells, thinking of natural transformations. That certainly makes a 
tremendous difference with Susan Niefield's notation who uses the 
converse convention and amply justifies Marco Grandis in giving 
references dated 1994 and 1996, i.e. more than 25 years posterior to my 
original paper.

With best regards

>
> You can find the strict version of that result in Prop. 1.4 of
>
>  - M. Grandis, Homotopical algebra in homotopical categories, Appl. 
> Categ.
>  Structures 2 (1994), 351-406.
>
>  I do not know if it has been written down elsewhere.
>
>  For sure, whiskering of natural transformations with functors is used 
> in:
>
>  - R. Street, Categorical structures, in: Handbook of Algebra, Vol. 1, 
> 529-577,
> North Holland, Amsterdam 1996.
>
>  where you can find the notion of a sesqui-category (which does not 
> assume the
>  "reduced interchange axiom" you are mentioning).
>
>  With best regards
>
>  M. Grandis
>
>>
>> Does anyone know of a reference for the following definition of a
>> bicategory?  The primitive composites are:
>>
>>   gf for composable 1-cells
>>   GF for vertically composable 2-cells
>>   f*G and F*g for horizontally composable pairs of each
>>
>> with appropriate axioms including (G*f')(g*F)=(g'*F)(G*f), for
>> F:f->f':X->Y and G:g->g':Y->Z.  The horizontal composite G*F is 
>> defined to
>> be the common value of the two vertical composites.
>>
>> -Susan
>
>
>

Re: horizontal composition

[Marco Grandis]

Dear Jean (and dear colleagues),

Everyone knows you defined bicategories (and Charles Ehresmann 2-
categories). The point of those two messages is that one can
equivalently define a 2-category or a bicategory using the following
primitive operations (and suitable axioms):

- the vertical composition of cells,
- the whisker composition of cells with maps (instead of the
horizontal composition of cells).

  (Which is precisely what we concretely do in Cat, when we define
horizontal composition of natural transformations: we use vertical
composition and whiskering, showing that the two possible ways of
defining horizontal composition give the same result, by the relevant
part of the middle-four interchange axiom - which I was calling
"reduced interchange".)

All this has some importance in homotopy, which is why I was
interested in it. For instance, take chain complexes with their
homotopies: then the vertical composition of homotopies is (strictly)
associative, whiskering is also associative (in the appropriate
sense), but reduced interchange fails and you do not have a
horizontal composition of homotopies. Such a structure is a sesqui-
category in Ross Street's sense - actually one might say "sesqui-
groupoid".

With best regards

Marco

Re: horizontal composition

[grandis]

You can find the strict version of that result in Prop. 1.4 of

 - M. Grandis, Homotopical algebra in homotopical categories, Appl. Categ.
 Structures 2 (1994), 351-406.

 I do not know if it has been written down elsewhere.

 For sure, whiskering of natural transformations with functors is used in:

 - R. Street, Categorical structures, in: Handbook of Algebra, Vol. 1, 529-577,
North Holland, Amsterdam 1996.

 where you can find the notion of a sesqui-category (which does not assume the
 "reduced interchange axiom" you are mentioning).

 With best regards

 M. Grandis

 >
 > Does anyone know of a reference for the following definition of a
 > bicategory?  The primitive composites are:
 >
 >   gf for composable 1-cells
 >   GF for vertically composable 2-cells
 >   f*G and F*g for horizontally composable pairs of each
 >
 > with appropriate axioms including (G*f')(g*F)=(g'*F)(G*f), for
 > F:f->f':X->Y and G:g->g':Y->Z.  The horizontal composite G*F is defined to
 > be the common value of the two vertical composites.
 >
 > -Susan

horizontal composition

[Susan Niefield]

Does anyone know of a reference for the following definition of a
bicategory?  The primitive composites are:

  gf for composable 1-cells
  GF for vertically composable 2-cells
  f*G and F*g for horizontally composable pairs of each

with appropriate axioms including (G*f')(g*F)=(g'*F)(G*f), for
F:f->f':X->Y and G:g->g':Y->Z.  The horizontal composite G*F is defined to
be the common value of the two vertical composites.

-Susan